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When studying polynomial or rational maps in one complex variable, the easiest to understand are those with connected Julia set, and with the property that the orbit of every critical point converges towards an attracting periodic orbit. Any connected component of such maps, within the space of suitably normalized maps of fixed degree n ? 2, will be called a hyperbolic component. John Milnor of Stony Brook University suggested there is a basic dichotomy -- that the topological boundary of such a hyperbolic component must be either semi-algebraic, defined by polynomial equalities and inequalities, or else not locally connected -- in a talk at the Bill Thurston Legacy Conference, June 23, 2014.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.